Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F

Average Error: 6.2 → 1.5

Time: 14.4s

Precision: binary64

Cost: 7236

Math

\[x - \frac{y \cdot \left(z - t\right)}{a} \]

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -2.2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- x (/ (* y (- z t)) a)))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))))
   (if (<= t_1 -2.2e+80)
     (fma y (/ (- t z) a) x)
     (if (<= t_1 2e+111)
       (+ x (/ (* y (- t z)) a))
       (- x (/ y (/ a (- z t))))))))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y * (z - t)) / a);
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -2.2e+80) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 2e+111) {
		tmp = x + ((y * (t - z)) / a);
	} else {
		tmp = x - (y / (a / (z - t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y * Float64(z - t)) / a))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= -2.2e+80)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 2e+111)
		tmp = Float64(x + Float64(Float64(y * Float64(t - z)) / a));
	else
		tmp = Float64(x - Float64(y / Float64(a / Float64(z - t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2.2e+80], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+111], N[(x + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.2
Target	0.7
Herbie	1.5

\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -2.20000000000000003e80

   1. Initial program 14.9

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Simplified 3.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)} \]
      Proof

      [Start] 14.9	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      sub-neg [=>] 14.9	\[ \color{blue}{x + \left(-\frac{y \cdot \left(z - t\right)}{a}\right)} \]
      +-commutative [=>] 14.9	\[ \color{blue}{\left(-\frac{y \cdot \left(z - t\right)}{a}\right) + x} \]
      *-commutative [=>] 14.9	\[ \left(-\frac{\color{blue}{\left(z - t\right) \cdot y}}{a}\right) + x \]
      associate-/l* [=>] 2.4	\[ \left(-\color{blue}{\frac{z - t}{\frac{a}{y}}}\right) + x \]
      distribute-neg-frac [=>] 2.4	\[ \color{blue}{\frac{-\left(z - t\right)}{\frac{a}{y}}} + x \]
      associate-/r/ [=>] 3.9	\[ \color{blue}{\frac{-\left(z - t\right)}{a} \cdot y} + x \]
      *-commutative [=>] 3.9	\[ \color{blue}{y \cdot \frac{-\left(z - t\right)}{a}} + x \]
      fma-def [=>] 3.9	\[ \color{blue}{\mathsf{fma}\left(y, \frac{-\left(z - t\right)}{a}, x\right)} \]
      sub-neg [=>] 3.9	\[ \mathsf{fma}\left(y, \frac{-\color{blue}{\left(z + \left(-t\right)\right)}}{a}, x\right) \]
      distribute-neg-in [=>] 3.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-z\right) + \left(-\left(-t\right)\right)}}{a}, x\right) \]
      +-commutative [=>] 3.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-\left(-t\right)\right) + \left(-z\right)}}{a}, x\right) \]
      remove-double-neg [=>] 3.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t} + \left(-z\right)}{a}, x\right) \]
      sub-neg [<=] 3.9	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t - z}}{a}, x\right) \]

   if -2.20000000000000003e80 < (*.f64 y (-.f64 z t)) < 1.99999999999999991e111

   1. Initial program 0.5

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   if 1.99999999999999991e111 < (*.f64 y (-.f64 z t))

   1. Initial program 18.4

      \[x - \frac{y \cdot \left(z - t\right)}{a} \]

   2. Simplified 2.5

      \[\leadsto \color{blue}{x - \frac{y}{\frac{a}{z - t}}} \]
      Proof

      [Start] 18.4	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      associate-/l* [=>] 2.5	\[ x - \color{blue}{\frac{y}{\frac{a}{z - t}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -2.2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 2 \cdot 10^{+111}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	33.2
Cost	1376

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-z\right)\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{+132}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-281}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-247}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-120}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 54000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+136}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+187}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	33.1
Cost	1376

\[\begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+133}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -4 \cdot 10^{+89}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-281}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-247}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-z\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-120}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 126000000:\\ \;\;\;\;\frac{z}{\frac{-a}{y}}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+135}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+186}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	33.2
Cost	1376

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+133}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{+89}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-281}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-247}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-z\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-120}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 126000000:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{a}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+137}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+187}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	22.1
Cost	1373

\[\begin{array}{l} t_1 := y \cdot \frac{t - z}{a}\\ \mathbf{if}\;x \leq -28000000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-163}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-89} \lor \neg \left(x \leq 95000000000\right) \land x \leq 7 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	0.8
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+283}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+111}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 6
Error	30.8
Cost	849

\[\begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+105}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{+89} \lor \neg \left(t \leq 1.6 \cdot 10^{+137}\right) \land t \leq 5.1 \cdot 10^{+185}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	30.8
Cost	848

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+133}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{+89}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+137}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+187}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	30.8
Cost	848

\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{+132}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+135}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+188}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	2.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+23} \lor \neg \left(z \leq 4 \cdot 10^{+18}\right):\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 10
Error	16.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+60} \lor \neg \left(y \leq 7.6 \cdot 10^{+113}\right):\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \end{array} \]

Alternative 11
Error	10.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-36} \lor \neg \left(z \leq 3.6 \cdot 10^{-41}\right):\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \end{array} \]

Alternative 12
Error	28.0
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-157}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-104}:\\ \;\;\;\;y \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	2.4
Cost	576

\[x + \frac{y}{a} \cdot \left(t - z\right) \]

Alternative 14
Error	30.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022354 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

  (- x (/ (* y (- z t)) a)))